Periods of automorphic forms, poles of \(L\)-functions and functorial lifting
In this paper the authors survey their impressive work linking four topics: poles or special values of certain Langlands \(L\)-functions, periods integrals involving automorphic forms, residual automorphic representations, and Langlands functorial transfer. In doing so, they emphasize commonalities in the many different objects that appear in treating these topics. This article will be useful to anyone wishing an introduction to this area of study. The paper begins with a brief review of automorphic forms and \(L\)-functions. The authors then illustrate their theme with an example: if \(\pi\) is an irreducible automorphic representation for \(GL_{2n}({\mathbb{A}})\) (where \({\mathbb{A}}\) is the ring of adeles of a number field \(F\)), then the following are equivalent. (i) The (partial) exterior square \(L\)-function for \(\pi\) has a simple pole at \(s=1\); (ii) a certain period integral (the Shalika integral) is nonzero for some choice of data; (iii) an automorphic representation \(\mathcal{E}_\pi\) on the split special orthogonal group \(SO_{4n}({\mathbb{A}})\), generated by the residues of certain Eisenstein series constructed using \(\pi\), is nonzero; (iv) the representation \(\pi\) is the Langlands functorial transfer of an irreducible cuspidal generic automorphic representation of the split special orthogonal group \(SO_{2n+1}({\mathbb{A}})\). The proof of this result is sketched. The authors then turn to more general situations, and explain that a similar set of equivalences should hold more widely. They explain the general philosophy and give many examples where such an equivalence can be deduced. The proof of the equivalence is typically sketched and careful references to the literature (many of them papers by the authors) are provided. They first present examples involving tensor product \(L\)-functions for classical groups. This is followed by a treatment of certain other \(L\)-functions for classical groups. Last, they treat constructions involving the exceptional groups. They also indicate some related open questions and conjectures.
scientific article; zbMATH DE number 5857682
scientific article